Rayleigh-Benard convection

For the description of modulated roll patterns away from threshold (possibly only slightly) the ampUtude B must be included, now also transformed to real space. A uniform scaling in is then no more possible and coupling terms like AQyB appear in Eq. 36 as well as derivative terms in the cubic non-linearities. The gradient expansion starting from Eqs. 32 and 33 is systematically truncated in such a way that all contributions to the growth rate (Eq. 34) are included [24, 83]. The additional equation for B is of the form 

The terms occurring are those allowed by symmetry. Due to the anisotropy more terms appear than in isotropic systems [94]. The clue for the characteristic appearance of the zigzag instabihty as a secondary bifurcation is that 4 is typically negative in nematics [23, 24] leading, in contrast to isotropic fluids, to amplification of transverse modulations of roll patterns. Model calculations that include this feature [25, 26] were quite successful in describing quahtatively the secondary instability and the behaviour beyond. 

In the following two Sections we will discuss and compare theoretical and experimental studies on RBC and EHC. We often use non-dimensionalized units. Thus we write wave numbers as qi = q/n/d, where the prime is sometimes omitted, and magnetic fields as Hi = hxHf with the splay Freedericksz transition field H/ = (n/d)[ku/ fioXa)] - Other quantities have been introduced before. Note that for the Cartesian components of the wave-vector we use two symbols, q = (qx, qy) = q,p)- 

Weakly non-linear theory predicts two tricritical points (TP) with a subcritical bifurcation in the field range 5 hx 26. Note that the upper TP at hx = 26 is slightly above the lower LP. One may expect rather complex non-linear behaviour in that range, which has not been worked out in detail for 5CB so far. The situation should be simpler for MBBA where the TPs (hx = 4.15 and 31) are below and well-separated from the Lifshitz points (hx = 36 and 62). 

The non-linear scenarios have not been systematically studied in experiments so far. Actually investigations at small fields hx lower TP) would be very interesting. According to the theory the upper limitation of the stability regime is now of the SV-type, beyond which the system cannot evade into stable oblique rolls and complex behaviour seems unavoidable. 

---

The same / lias also boon used as a model for spatiotemporal interniittency in Rayleigh-Benard convection ([cili88], [davl89]). 

W. J. Goux, L. A. Verkruyse, S. J. Salter 1990, (The impact of Rayleigh-Benard convection on NMR pulsed-field-gra-dient diffusion measurements), J. Mag. Reson. 88, 609. 

V. Croquette, P. Le Gal, A. Pocheau, and R. Guglielmetti, Large-scale characterization in a Rayleigh-Benard convective pattern, Europhys. Lett., 1, 393-399 (1986). 

In this framework an interesting example is the Lagrangian motion in velocity field given by a simple model for Rayleigh-Benard convection [31], which is given by the stream function ... 

Plapp B. R, Egolf D. A., Bodenschatz E. and Pesch W., Dynamics and Selection of Giant Spirals in Rayleigh-Benard Convection, Phys. Rev. Lett. 81, 5334 (1998). 

The first problem considered is the classic problem of Rayleigh-Benard convection -namely the instability that is due to buoyancy forces in a quiescent fluid layer that is heated... 

Problem 12-14. Rayleigh-Benard Convection - One Free and One Rigid Boundary. We... 

B.I. Shraiman. Diffusive transport in a Rayleigh-Benard convection cell. Phys. Rev. A, 36 261-267, 1987. 

T.H. Solomon and J.P. Gollub. Passive transport in steady Rayleigh-Benard convection. Phys. Fluids, 31 1372-1379, 1988. 

The most well-known example of pattern formation is Rayleigh-Benard convection which appears when a fluid layer is uniformly heated from below... 

If condition (29) is violated, more complicated patterns may appear. As an example, let us consider the Gertsberg-Sivashinsky equation that was derived in the problem of Rayleigh-Benard convection in a layer between weakly conducting boundaries [41],... 

We come to the conclusion that only the roll patterns inside the stability interval 0 < K < 1/ /3 are stable. The stability interval is also called the Busse balloon, for it was first discovered by Busse et al. in the context of the Rayleigh-Benard convection patterns [40]. See the diagram in Fig. 12. 

Spiral-defect chaos in Rayleigh-Benard convection. The most remarkable phenomenon that needs an extension of the Swift-Hohenberg model for its explanation, is the development of spiral-defect chaos in Rayleigh-Benard convection [56], [6], which involves rotating spirals, target patterns, dislocations etc. The origin of this complicated behavior is the creation of a two-dimensional mean flow... 

We have also discussed the formation of spatio-temporal patterns in non-variational systems. A typical example of such systems at nano-meter scales is reaction-diffusion systems that are ubiquitous in biology, chemical catalysis, electrochemistry, etc. These systems are characterized by the energy supply from the outside and can exhibit complex nonlinear behavior like oscillations and waves. A macroscopic example of such a system is Rayleigh-Benard convection accompanied by mean flow that leads to strong distortion of periodic patterns and the formation of labyrinth patterns and spiral waves. Similar nano-meter scale patterns are observed during phase separation of diblock copolymer Aims in the presence of hydrodynamic effects. The pattern s nonlinear dynamics in both macro- and nano-systems can be described by a Swift-Hohenberg equation coupled to the non-local mean-flow equation. 

Newell-Whitehead-Segel equation, 23 Non-potential effects, 41 Orientational instability, 283 Pattern formation, 1,11 Phase field, 168 Polymerization wave, 235, 239 Polymerization waves, 236, 238 Propagating front, 260-261 Quantum dots, 123-124 Rayleigh-Benard convection, 61 SHS, 247-248 Smectics, 57 Spiral wave, 47 Stochastic oscillations, 92 Stripes, 2, 10 Surface diffusion, 126... 

Consider the following systems of equations which represent a simplified model of Rayleigh-Benard convection ... 

Assenheimer, M., Steinberg, V. Observation of coexisting upflow and downflow hexagons in Boussinesq Rayleigh-Benard convection. Phys. Rev. Lett. 76, 756-759 (1996)... 

Convection instabilities in simple isotropic fluids, like Rayleigh-Benard convection (for a recent review see Bodensctiatz et are completely under-... 

E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Benard convection, Annu. Rev. Fluid Mech. 32, 709-778, (2000). 

The temperature gradient within the droplet perpendicular to the substrate can induce Rayleigh-Benard convection cells. The corresponding convective cells grow stronger... 

The presence of surfactants, which adsorb at the liquid-vapor interface and reduce the surface tension, can also have a large impact on the evaporation-driven pattern formation. Inhomogeneities in the surfactant distribution create a surface tension gradient and a corresponding (additional) Marangoni flow. With respect to the Rayleigh-Benard convection, it has been shown that the surfactant-driven flow can favor the formation of convection cells and considerably alter the deposition patterns [7]. 

The order parameter was arguably first introduced by Landau at the equilibrium thermodynamic level to study phase behavior. Order parameter models can also be motivated through classical bifurcation theory, and several physical systems have been recently modeled in this way. They include Rayleigh-Benard convection, Faraday waves,or pattern formation in optical systems. Close to a bifurcation point, these models asymptotically describe the system under study, but they are now routinely used, in a phenomenological feshion, to describe highly nonlinear phenomena. 

We present a brief introduction to coupled transport processes described macroscopically by hydrodynamic equations, the Navier-Stokes equations [4]. These are difficult, highly non-linear coupled partial differential equations they are frequently approximated. One such approximation consists of the Lorenz equations [5,6], which are obtained from the Navier-Stokes equations by Fourier transform of the spatial variables in those equations, retention of first order Fourier modes and restriction to small deviations from a bifurcation of an homogeneous motionless stationary state (a conductive state) to an inhomogeneous convective state in Rayleigh-Benard convection (see the next paragraph). The Lorenz equations have been applied successfully in various fields ranging from meteorology to laser physics. 

Pattern formation in hydrodynamic instabilities has been studied intensely over the last decades [1, 2], Although Rayleigh-Benard convection (RBC) in simple fluids has been the prime example [3], the rich variety of scenarios found in nematic hquid crystals (LCs) has attracted increased attention. 

---